a) = x^2 - 2x + 1 + 4y^2 + 4y + 1
= ( x - 1 )^2 + ( 2y + 1 )^2
b) = 4x^2 + 4x +1 + 4y^2 + 4y + 1
= ( 2x + 1 )^2 + ( 2y + 1 )^2
c) = 9x^2 - 12x + 4 + 16y^2 - 24y + 9
=( 3x - 2 )^2 + ( 4y - 3 )^2
d) = 4x^2 + 4xy+ y^2 + x^2 - 2xz + z^2
= ( 2x + y )^2 + ( x - z )^2
1. x2 - 2x + 2 + 4y2 + 4y
= x2 - 2x + 1 + 4y2 + 4y + 1
= ( x - 1 )2 + ( 2y + 1 )2
2. 4x2 + 4x + 4y2 + 4y + 2
= 4x2 + 4x + 1 + 4y2 + 4y + 1
= ( 2x + 1 )2 + ( 2y + 1 )2
3. -12x + 13 - 24y + 9x2 + 16y2
= 9x2 - 12x + 4 + 16y2 - 24y + 9
= ( 3x - 2 )2 + ( 4y - 3 )2
4. 5x2 + y2 + z2 + 4xy - 2xz
= 4x2 + 4xy + y2 + x2 - 2xz + z2
= ( 2x + y )2 + ( x - z )2